Normed and Banach spaces:
Definitions and examples for Banach spaces and bounded linear operators.
Examples include some `classical' Banach spaces $C(X)$, $\ell^p$, $c_0$,
$L^p([0,1])$ and more general $L^p$ examples ($1 \leq p \leq \infty$).
Use of series in Banach spaces (convergent or absolutely convergent),
basic concepts from Lebesgue integration. we show Hölder's and
Minkowski's inequalities (vesions for sums and integrals).
$\ell^p$ increases with $p$ while $L^p([0,1])$ decreases
and the inclusion maps as examples of operators.

Baire category theorem and some of its consequences (open mapping).
An application to Fourier series.